Square Number Sequence Index from Value Calculator

Enter Value

Output shows square status, exact index when applicable, nearest indices, and deltas from neighbor squares.

Results

Input V	Perfect Square?	Exact Index n	√V (decimal)	⌊√V⌋	⌈√V⌉	Lower Square	Upper Square	Δ to Lower	Δ to Upper
100	yes	10	10	10	10	100	100	0	0

Formula Used

For a square sequence value V, the index n satisfies n² = V with n = √V. A value is a perfect square if √V is an integer.

When V is not a perfect square, define n_floor = ⌊√V⌋ and n_ceil = ⌈√V⌉. The nearest neighbor squares are n_floor² and n_ceil², with deltas Δ_lower = V − n_floor² and Δ_upper = n_ceil² − V.

For large integers, the script optionally uses BCMath to compute an integer square root without floating-point rounding errors.

How to Use This Calculator

Enter a non-negative value V and click Compute.
See whether V is a perfect square and, if so, its index n.
When not a perfect square, use the floor and ceiling indices to identify the closest neighbor squares and their deltas.
Use the CSV or PDF buttons to export the results table for records.
Share your calculation with the link button for easy collaboration.

Example Data

V	Perfect Square?	Index n (if exact)	√V (decimal)	⌊√V⌋	⌈√V⌉	Lower Square	Upper Square
0	yes	0	0	0	0	0	0
1	yes	1	1	1	1	1	1
2	no	-	1.414213562373	1	2	1	4
3	no	-	1.732050807569	1	2	1	4
4	yes	2	2	2	2	4	4
5	no	-	2.2360679775	2	3	4	9
7	no	-	2.645751311065	2	3	4	9
8	no	-	2.828427124746	2	3	4	9
9	yes	3	3	3	3	9	9
10	no	-	3.162277660168	3	4	9	16
15	no	-	3.872983346207	3	4	9	16
16	yes	4	4	4	4	16	16
20	no	-	4.472135955	4	5	16	25
36	yes	6	6	6	6	36	36
49	yes	7	7	7	7	49	49
50	no	-	7.071067811865	7	8	49	64
81	yes	9	9	9	9	81	81
100	yes	10	10	10	10	100	100
121	yes	11	11	11	11	121	121

Click a value to try it quickly:

0 1 2 3 4 5 7 8 9 10 15 16 20 36 49 50 81 100 121

Example: Using Square Number Sequence Index from Value

Let V = 50. We want the index n such that n² = V if V is a perfect square, or determine the nearest indices otherwise.

Compute √V = √50 ≈ 7.07106781187.
Because √V is not an integer, 50 is not a perfect square.
Find ⌊√V⌋ = 7 and ⌈√V⌉ = 8.
Neighbor squares: 7² = 49 and 8² = 64.
Deltas: Δ_lower = 50 − 49 = 1, Δ_upper = 64 − 50 = 14.

Try this example

Common Squares and Indices (n, n²)

n	n²
0	0
1	1
2	4
3	9
4	16
5	25
6	36
7	49
8	64
9	81
10	100
11	121
12	144
13	169
14	196
15	225
16	256
17	289
18	324
19	361
20	400
21	441
22	484
23	529
24	576
25	625
26	676
27	729
28	784
29	841
30	900

Consecutive Square Gaps ( (n+1)² − n² = 2n+1 )

n	(n+1)²	n²	Gap 2n+1
0	1	0	1
1	4	1	3
2	9	4	5
3	16	9	7
4	25	16	9
5	36	25	11
6	49	36	13
7	64	49	15
8	81	64	17
9	100	81	19
10	121	100	21
11	144	121	23
12	169	144	25
13	196	169	27
14	225	196	29
15	256	225	31
16	289	256	33
17	324	289	35
18	361	324	37
19	400	361	39
20	441	400	41
21	484	441	43
22	529	484	45
23	576	529	47
24	625	576	49
25	676	625	51
26	729	676	53
27	784	729	55
28	841	784	57
29	900	841	59
30	961	900	61

The distance between consecutive squares grows linearly with n.

Index Range by Value Interval (V ∈ [n², (n+1)²))

n	Interval for V	Size	Interpretation
0	[0, 1)	1	For V in this range, ⌊√V⌋ = 0
1	[1, 4)	3	For V in this range, ⌊√V⌋ = 1
2	[4, 9)	5	For V in this range, ⌊√V⌋ = 2
3	[9, 16)	7	For V in this range, ⌊√V⌋ = 3
4	[16, 25)	9	For V in this range, ⌊√V⌋ = 4
5	[25, 36)	11	For V in this range, ⌊√V⌋ = 5
6	[36, 49)	13	For V in this range, ⌊√V⌋ = 6
7	[49, 64)	15	For V in this range, ⌊√V⌋ = 7
8	[64, 81)	17	For V in this range, ⌊√V⌋ = 8
9	[81, 100)	19	For V in this range, ⌊√V⌋ = 9
10	[100, 121)	21	For V in this range, ⌊√V⌋ = 10
11	[121, 144)	23	For V in this range, ⌊√V⌋ = 11
12	[144, 169)	25	For V in this range, ⌊√V⌋ = 12
13	[169, 196)	27	For V in this range, ⌊√V⌋ = 13
14	[196, 225)	29	For V in this range, ⌊√V⌋ = 14
15	[225, 256)	31	For V in this range, ⌊√V⌋ = 15

n	n²
0	0
1	1
2	4
3	9
4	16
5	25
6	36
7	49
8	64
9	81
10	100
11	121
12	144
13	169
14	196
15	225
16	256
17	289
18	324
19	361
20	400
21	441
22	484
23	529
24	576
25	625
26	676
27	729
28	784
29	841
30	900

n	(n+1)²	n²	Gap 2n+1
0	1	0	1
1	4	1	3
2	9	4	5
3	16	9	7
4	25	16	9
5	36	25	11
6	49	36	13
7	64	49	15
8	81	64	17
9	100	81	19
10	121	100	21
11	144	121	23
12	169	144	25
13	196	169	27
14	225	196	29
15	256	225	31
16	289	256	33
17	324	289	35
18	361	324	37
19	400	361	39
20	441	400	41
21	484	441	43
22	529	484	45
23	576	529	47
24	625	576	49
25	676	625	51
26	729	676	53
27	784	729	55
28	841	784	57
29	900	841	59
30	961	900	61

n	n²
0	0
1	1
2	4
3	9
4	16
5	25
6	36
7	49
8	64
9	81
10	100
11	121
12	144
13	169
14	196
15	225
16	256
17	289
18	324
19	361
20	400
21	441
22	484
23	529
24	576
25	625
26	676
27	729
28	784
29	841
30	900

n	(n+1)²	n²	Gap 2n+1
0	1	0	1
1	4	1	3
2	9	4	5
3	16	9	7
4	25	16	9
5	36	25	11
6	49	36	13
7	64	49	15
8	81	64	17
9	100	81	19
10	121	100	21
11	144	121	23
12	169	144	25
13	196	169	27
14	225	196	29
15	256	225	31
16	289	256	33
17	324	289	35
18	361	324	37
19	400	361	39
20	441	400	41
21	484	441	43
22	529	484	45
23	576	529	47
24	625	576	49
25	676	625	51
26	729	676	53
27	784	729	55
28	841	784	57
29	900	841	59
30	961	900	61

Related Calculators

n	n²
0	0
1	1
2	4
3	9
4	16
5	25
6	36
7	49
8	64
9	81
10	100
11	121
12	144
13	169
14	196
15	225
16	256
17	289
18	324
19	361
20	400
21	441
22	484
23	529
24	576
25	625
26	676
27	729
28	784
29	841
30	900

n	(n+1)²	n²	Gap 2n+1
0	1	0	1
1	4	1	3
2	9	4	5
3	16	9	7
4	25	16	9
5	36	25	11
6	49	36	13
7	64	49	15
8	81	64	17
9	100	81	19
10	121	100	21
11	144	121	23
12	169	144	25
13	196	169	27
14	225	196	29
15	256	225	31
16	289	256	33
17	324	289	35
18	361	324	37
19	400	361	39
20	441	400	41
21	484	441	43
22	529	484	45
23	576	529	47
24	625	576	49
25	676	625	51
26	729	676	53
27	784	729	55
28	841	784	57
29	900	841	59
30	961	900	61